Related papers: Runge-Kutta and Networks
We study the connectedness locus $\mathcal{N}$ for the family of iterated function systems of pairs of homogeneous affine-linear maps in the plane. We prove this set is regular closed (i.e., it is the closure of its interior) away from the…
We introduce economical versions of standard implicit ODE solvers that are specifically tailored for the efficient and accurate simulation of neural networks. These reformulations allow to achieve a significant increase in the efficiency of…
We consider arbitrary bounded discrete time series originating from dynamical system with recursivity. More precisely, we provide an explicit construction of recurrent neural networks which effectively approximate the corresponding discrete…
In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are…
A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under the exact solution of their governing PDEs. However, standard temporal schemes,…
Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components…
Classical convergence theory of Runge-Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic…
This article extends the theory of dual-consistent summation-by-parts (SBP) and generalized SBP (GSBP) time-marching methods by showing that they are implicit Runge-Kutta schemes. Through this connection, the accuracy theory for the…
The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a…
We propose a kernel compression method for solving Distributed-Order (DO) Fractional Partial Differential Equations (DOFPDEs) at the cost of solving corresponding local-in-time PDEs. The key concepts are (1) discretization of the integral…
This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable…
Many control, optimization, and learning algorithms rely on discretizations of continuous-time contracting systems, where preservation of contractivity under numerical integration is key for stability, robustness, and reliable fixed-point…
Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires…
With the increasing industrial demands, two families of high-order numerical schemes are widely used within the computational fluid dynamics community. One is the method of line, which relies on Runge-Kutta (RK) time-stepping applied to a…
Runge--Kutta (RK) methods are widely used techniques for solving a class of initial value problems. In this article, we introduce an adaptive multiquadratic (MQ) radial basis function (RBF)-based method to develop enhanced explicit RK…
The cubic spline interpolation method, the Runge--Kutta method, and the Newton-Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated…
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal…
The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space…